Question

1.2
the table shows some values of a function. on which intervals is the functions average rate of change positive? select all that apply.

x	0	1	2	3
f(x)	50	75	40	65

a. from x = 0 to x = 1
c. from x = 0 to x = 3
e. from x = 1 to x = 3
b. from x = 0 to x = 2
d. from x = 1 to x = 2
f. from x = 2 to x = 3

Explanation:

Step1: Recall Average Rate of Change Formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). A positive average rate of change occurs when \( f(b) - f(a) > 0 \) (since \( b - a > 0 \) for \( b > a \)).

Step2: Analyze Interval a: \( x = 0 \) to \( x = 1 \)

\( f(0) = 50 \), \( f(1) = 75 \).
Average rate of change: \(\frac{75 - 50}{1 - 0} = \frac{25}{1} = 25\).
Since \( 25 > 0 \), the rate of change is positive.

Step3: Analyze Interval b: \( x = 0 \) to \( x = 2 \)

\( f(0) = 50 \), \( f(2) = 40 \).
Average rate of change: \(\frac{40 - 50}{2 - 0} = \frac{-10}{2} = -5\).
Since \( -5 < 0 \), the rate of change is negative.

Step4: Analyze Interval c: \( x = 0 \) to \( x = 3 \)

\( f(0) = 50 \), \( f(3) = 65 \).
Average rate of change: \(\frac{65 - 50}{3 - 0} = \frac{15}{3} = 5\).
Since \( 5 > 0 \), the rate of change is positive.

Step5: Analyze Interval d: \( x = 1 \) to \( x = 2 \)

\( f(1) = 75 \), \( f(2) = 40 \).
Average rate of change: \(\frac{40 - 75}{2 - 1} = \frac{-35}{1} = -35\).
Since \( -35 < 0 \), the rate of change is negative.

Step6: Analyze Interval e: \( x = 1 \) to \( x = 3 \)

\( f(1) = 75 \), \( f(3) = 65 \).
Average rate of change: \(\frac{65 - 75}{3 - 1} = \frac{-10}{2} = -5\).
Since \( -5 < 0 \), the rate of change is negative.

Step7: Analyze Interval f: \( x = 2 \) to \( x = 3 \)

\( f(2) = 40 \), \( f(3) = 65 \).
Average rate of change: \(\frac{65 - 40}{3 - 2} = \frac{25}{1} = 25\).
Since \( 25 > 0 \), the rate of change is positive.

Answer:

a. From \( x = 0 \) to \( x = 1 \), c. From \( x = 0 \) to \( x = 3 \), f. From \( x = 2 \) to \( x = 3 \)